Linear Model Selection and Regularization

Advance Analytics with R (UG 21-24)

Ayush Patel

Before we start

Please load the following packages

library(tidyverse)
library(MASS)
library(ISLR)
library(ISLR2)
library(nnet)### get this if you don't
library(e1071) ## get this if you don't
library(modeldata)## get this if you don't



Access lecture slide from bit.ly/aar-ug

Warrior's armor(gusoku)
Source: Armor (Gusoku)

Hello

I am Ayush.

I am a researcher working at the intersection of data, law, development and economics.

I teach Data Science using R at Gokhale Institute of Politics and Economics

I am a RStudio (Posit) certified tidyverse Instructor.

I am a Researcher at Oxford Poverty and Human development Initiative (OPHI), at the University of Oxford.

Reach me

ayush.ap58@gmail.com

ayush.patel@gipe.ac.in

Why are we still talkig about linear models?

When there are many fancy things to try out!!

  • Inference Advantage
  • Real World Problems Advantage
  • Competitive compared to non-linear models.

Therefore, we discuss linear models beyond least squares estimate.

What do we gain?

Prediction Accuracy:

  • If Y and X are approximately linear, least square approach has low bias.
  • For \(n>>p\), least square has low variance as well.
  • But, if \(n>>p\) is not held, least square has high variance and poor prediction ability on future observations.
  • And if \(p>n\), then there are infinitely many solutions to least square method.

What do we gain?

Model Interpretability

  • too many predictors make a complex model.
  • Sometimes, some predictors may not even have a real relation with the response.
  • We reduce coeffs of Irrelevant variables to 0.
  • We discuss approaches of feature selection.

Subset Selection

“This approach involves identifying a subset of the p predictors that we believe to be related to the response. We then fit a model using least squares on the reduced set of variables.”

  • Best Subset
  • Stepwise selection

Subset Selection - Best Subset

A least square model is fit for all possible combinations of p predictors.

So, if there are 3 predictors (\(p_1,p_2,p_3\)), we fit the following models:

model 1 \(y = \beta_a + \beta_1*p_1\)
model 2 \(y = \beta_b + \beta_2*p_2\)
model 3 \(y = \beta_c + \beta_3*p_3\)
.
.
model 7 \(y = \beta_0 + \beta_e*p_1 + \beta_f*p_2 + \beta_g*p_3\)

Select the best from these

Subset Selection - Best Subset

  • A null model is defined with no predictors. Name it \(M_0\)

  • For each \(k\), where \(k=1,2,3..p\), select the best model from the all \((_k^p)\) combinations. Use RSS or \(R^2\). Call it \(M_k\)

  • Select a single best model from \(M_0, M_1,.....,M_p\). Use prediction error on a validation set,\(C_p\), AIC, BIC, adjusted \(R^2\) or use the cross validation method.

Subset Selection - Best Subset

Issues:

  • Computational limitations.
  • ” The larger the search space, the higher the chance of finding models that look good on the training data, even though they might not have any predictive power on future data. “

Subset Selection - Forward Stepwise Selection

  • Start with Null model. \(M_0\)
  • For all \(p-k\) models, choose the best using \(R^2\) or RSS.
  • Select a single best model from \(M_0, M_1,.....,M_p\). Use prediction error on a validation set,\(C_p\), AIC, BIC, adjusted \(R^2\) or use the cross validation method.

Subset Selection - Forward Stepwise Selection

Issues:

  • “Though forward stepwise tends to do well in practice, it is not guaranteed to find the best possible model out of all 2p models containing subsets of the p predictors.”

Subset Selection - Backward Stepwise Selection

  • Start with full model. \(M_p\)
  • Consider all k models that contain all but one of the predictors in \(M_k\), for a total of k − 1 predictors. Choose the best among these k models, and call it Mk−1. Here best is defined as having smallest RSS or highest \(R^2\)
  • Select a single best model from \(M_0, M_1,.....,M_p\). Use prediction error on a validation set,\(C_p\), AIC, BIC, adjusted \(R^2\) or use the cross validation method.

Choose wisely

We generate several models using best subset selection, forward stepwise and backward stepwise.

We must choose carefully which models to pick. We ideally want a model that will have the lowest test error rate.

  • Estimate test error rate by adjusting training error rate for bias due to overfitting.
  • Cross validation.

The four horsemen

\(C_p\), AIC,BIC, Adjusted \(R^2\)

Recall that \(MSE = RSS/n\). Also, since we use least squares approach to choose coeffs in a manner where training RSS is minimized, we end up getting a training MSE is an underestimate of test MSE.

That is the reason \(R^2\) and training RSS are not suitable for selecting from many models.

Therefore we learn about \(C_p\), Akaike Information criterion (AIC), Bayesian Information Criterion (BIC) and Adjusted \(R^2\)

The four horsemen - \(C_p\)

Say there is a least square model with \(d\) predictors.

The estimate of test error rate is given by:

\[C_p = \frac{1}{n}(RSS + 2d\hat\sigma^2)\]

\(\hat\sigma^2\) is the estimate of variance of \(\epsilon\), estimated using a full model.

The four horsemen - AIC

AIC is defined for a range of maximum likelihood models.

\[AIC = 2d - 2ln(L)\]

\[AIC = \frac{1}{n}(RSS + 2d\hat\sigma^2)\]

The four horsemen - BIC

BIC, for a least square model with d predictors:

\[BIC = \frac{1}{n}(RSS + log(n)d\hat\sigma^2 )\]

The four horsemen - Adjusted \(R^2\)

\[Adjusted\hspace{1mm} R^2 = \frac{RSS(n-d-1)}{TSS(n-1)}\]

Cross Validation - an alternative to the horsemen

  • Instead of adjusting we can directly estimate test error rate.
  • Makes fewer assumptions about underlying model.

Exercise - best subset

See 6.5.1 in ISLR

Apply best subset approach for the Auto data in ISLR2 package.

Use mpg as response.

What is the best model? Feel free to use ggplot for charts and broom to collect stats from model object.

Exercise - Validation set

See 6.5.1 in ISLR

Apply validation set approach on Auto data in ISLR2 package.

Use mpg as response.

Which is the best model?

Shrinkage Methods

In subsetset selection methods we are working with least squares.

Alternatively, we try to reduce the coefficients in a full model to zero. This is referred to shrinkage, constraining or regularization.

This reduces variance of coefficients, leading to a better bit.

Ridge and Lasso are two techniques that allow us to do this.

Ridge Regression

In least square method we minimize RSS to estimate coefficients.

In Ridge we minimize:

\[RSS + \lambda \sum_{j=1}^p\beta_j^2\]

\(\lambda >= 0\) is the tuning parameter.

Ridge Regression produces mroe than 1 set of coefficients.
Ridge Regression is not scale equivariant.

Ridge Regression

Since it is not scale equivariant, apply only after standardizing the predictors.

\[\tilde x = \frac{x_{ij}}{\sqrt{\frac{1}{n}\sum_{i=1}^j(x_{ij} - \bar{x_{ij}})^2}}\]

Apply Ridge

## Remove all NA

Hitters |>
  na.omit() -> hit_no_miss

## Model Matrix

x <- model.matrix(Salary ~., hit_no_miss)[,-1]

y <- hit_no_miss$Salary

## ridge

10^seq(10, -2, length = 100) -> lambdas

glmnet(x, y, alpha = 0, 
       lambda = lambdas) -> mod_ridge

## Model Object

mod_ridge|>
  broom::tidy()|>
  knitr::kable()%>%
  kableExtra::kable_styling()%>%
  kableExtra::scroll_box(width = "100%", height = "400px")
term step estimate lambda dev.ratio
(Intercept) 1 535.9256935 1.000000e+10 0.0000003
(Intercept) 2 535.9256328 7.564633e+09 0.0000004
(Intercept) 3 535.9255525 5.722368e+09 0.0000005
(Intercept) 4 535.9254464 4.328761e+09 0.0000006
(Intercept) 5 535.9253061 3.274549e+09 0.0000008
(Intercept) 6 535.9251207 2.477076e+09 0.0000011
(Intercept) 7 535.9248756 1.873817e+09 0.0000015
(Intercept) 8 535.9245515 1.417474e+09 0.0000019
(Intercept) 9 535.9241232 1.072267e+09 0.0000026
(Intercept) 10 535.9235569 8.111308e+08 0.0000034
(Intercept) 11 535.9228083 6.135907e+08 0.0000045
(Intercept) 12 535.9218187 4.641589e+08 0.0000059
(Intercept) 13 535.9205105 3.511192e+08 0.0000079
(Intercept) 14 535.9187812 2.656088e+08 0.0000104
(Intercept) 15 535.9164952 2.009233e+08 0.0000137
(Intercept) 16 535.9134732 1.519911e+08 0.0000181
(Intercept) 17 535.9094783 1.149757e+08 0.0000240
(Intercept) 18 535.9041974 8.697490e+07 0.0000317
(Intercept) 19 535.8972165 6.579332e+07 0.0000419
(Intercept) 20 535.8879884 4.977024e+07 0.0000554
(Intercept) 21 535.8757898 3.764936e+07 0.0000732
(Intercept) 22 535.8596646 2.848036e+07 0.0000968
(Intercept) 23 535.8383491 2.154435e+07 0.0001279
(Intercept) 24 535.8101735 1.629751e+07 0.0001691
(Intercept) 25 535.7729305 1.232847e+07 0.0002236
(Intercept) 26 535.7237037 9.326033e+06 0.0002955
(Intercept) 27 535.6586397 7.054802e+06 0.0003906
(Intercept) 28 535.5726481 5.336699e+06 0.0005162
(Intercept) 29 535.4590057 4.037017e+06 0.0006822
(Intercept) 30 535.3088355 3.053856e+06 0.0009015
(Intercept) 31 535.1104212 2.310130e+06 0.0011912
(Intercept) 32 534.8483072 1.747528e+06 0.0015737
(Intercept) 33 534.5021185 1.321941e+06 0.0020786
(Intercept) 34 534.0450197 1.000000e+06 0.0027448
(Intercept) 35 533.4417075 7.564633e+05 0.0036233
(Intercept) 36 532.6458129 5.722368e+05 0.0047807
(Intercept) 37 531.5965583 4.328761e+05 0.0063041
(Intercept) 38 530.2145014 3.274549e+05 0.0083062
(Intercept) 39 528.3961842 2.477076e+05 0.0109327
(Intercept) 40 526.0283002 1.873817e+05 0.0143452
(Intercept) 41 522.9117270 1.417474e+05 0.0188114
(Intercept) 42 518.8425089 1.072267e+05 0.0246060
(Intercept) 43 513.5499429 8.111308e+04 0.0320807
(Intercept) 44 506.7007228 6.135907e+04 0.0416500
(Intercept) 45 497.8943274 4.641589e+04 0.0537819
(Intercept) 46 486.6653876 3.511192e+04 0.0689717
(Intercept) 47 472.4983784 2.656088e+04 0.0876909
(Intercept) 48 454.8612601 2.009233e+04 0.1103053
(Intercept) 49 433.2682588 1.519911e+04 0.1369603
(Intercept) 50 407.3560502 1.149757e+04 0.1674595
(Intercept) 51 377.0065699 8.697490e+03 0.2011472
(Intercept) 52 342.4514686 6.579332e+03 0.2368793
(Intercept) 53 304.3628598 4.977024e+03 0.2731090
(Intercept) 54 263.8466546 3.764936e+03 0.3081306
(Intercept) 55 222.3783088 2.848036e+03 0.3403876
(Intercept) 56 181.6206640 2.154435e+03 0.3687727
(Intercept) 57 143.2209084 1.629751e+03 0.3927800
(Intercept) 58 108.6007107 1.232847e+03 0.4125082
(Intercept) 59 78.7760218 9.326033e+02 0.4284125
(Intercept) 60 54.3251995 7.054802e+02 0.4412059
(Intercept) 61 35.4632735 5.336699e+02 0.4515636
(Intercept) 62 22.0716345 4.037017e+02 0.4601078
(Intercept) 63 13.6996547 3.053856e+02 0.4673437
(Intercept) 64 10.0897901 2.310130e+02 0.4737111
(Intercept) 65 10.3680866 1.747528e+02 0.4794882
(Intercept) 66 14.2600256 1.321941e+02 0.4849761
(Intercept) 67 21.2474898 1.000000e+02 0.4903658
(Intercept) 68 30.6934829 7.564633e+01 0.4957413
(Intercept) 69 42.0855152 5.722368e+01 0.5011443
(Intercept) 70 54.9738422 4.328761e+01 0.5065677
(Intercept) 71 68.6872524 3.274549e+01 0.5119093
(Intercept) 72 82.6164012 2.477076e+01 0.5170544
(Intercept) 73 96.3405543 1.873817e+01 0.5219007
(Intercept) 74 109.2829120 1.417474e+01 0.5263263
(Intercept) 75 121.0394128 1.072267e+01 0.5302523
(Intercept) 76 131.3383871 8.111308e+00 0.5336375
(Intercept) 77 140.0270464 6.135907e+00 0.5364782
(Intercept) 78 147.0439149 4.641589e+00 0.5388070
(Intercept) 79 152.5289391 3.511192e+00 0.5406688
(Intercept) 80 156.6073700 2.656088e+00 0.5421321
(Intercept) 81 159.6160967 2.009233e+00 0.5432400
(Intercept) 82 161.6138012 1.519911e+00 0.5440843
(Intercept) 83 162.9350244 1.149757e+00 0.5446980
(Intercept) 84 163.6938252 8.697490e-01 0.5451444
(Intercept) 85 164.1218504 6.579332e-01 0.5454543
(Intercept) 86 164.3266996 4.977024e-01 0.5456662
(Intercept) 87 164.4197999 3.764936e-01 0.5458062
(Intercept) 88 164.4398691 2.848036e-01 0.5458983
(Intercept) 89 164.4099707 2.154435e-01 0.5459591
(Intercept) 90 164.3701706 1.629751e-01 0.5459979
(Intercept) 91 164.3247947 1.232847e-01 0.5460232
(Intercept) 92 164.2813513 9.326030e-02 0.5460397
(Intercept) 93 164.2457978 7.054800e-02 0.5460503
(Intercept) 94 164.2135548 5.336700e-02 0.5460580
(Intercept) 95 164.1869190 4.037020e-02 0.5460629
(Intercept) 96 164.1646430 3.053860e-02 0.5460666
(Intercept) 97 164.1460911 2.310130e-02 0.5460692
(Intercept) 98 164.1319561 1.747530e-02 0.5460709
(Intercept) 99 164.1204583 1.321940e-02 0.5460723
(Intercept) 100 164.1132161 1.000000e-02 0.5460731
AtBat 1 0.0000001 1.000000e+10 0.0000003
AtBat 2 0.0000001 7.564633e+09 0.0000004
AtBat 3 0.0000001 5.722368e+09 0.0000005
AtBat 4 0.0000001 4.328761e+09 0.0000006
AtBat 5 0.0000002 3.274549e+09 0.0000008
AtBat 6 0.0000002 2.477076e+09 0.0000011
AtBat 7 0.0000003 1.873817e+09 0.0000015
AtBat 8 0.0000004 1.417474e+09 0.0000019
AtBat 9 0.0000005 1.072267e+09 0.0000026
AtBat 10 0.0000007 8.111308e+08 0.0000034
AtBat 11 0.0000009 6.135907e+08 0.0000045
AtBat 12 0.0000012 4.641589e+08 0.0000059
AtBat 13 0.0000016 3.511192e+08 0.0000079
AtBat 14 0.0000020 2.656088e+08 0.0000104
AtBat 15 0.0000027 2.009233e+08 0.0000137
AtBat 16 0.0000036 1.519911e+08 0.0000181
AtBat 17 0.0000047 1.149757e+08 0.0000240
AtBat 18 0.0000063 8.697490e+07 0.0000317
AtBat 19 0.0000083 6.579332e+07 0.0000419
AtBat 20 0.0000109 4.977024e+07 0.0000554
AtBat 21 0.0000145 3.764936e+07 0.0000732
AtBat 22 0.0000191 2.848036e+07 0.0000968
AtBat 23 0.0000253 2.154435e+07 0.0001279
AtBat 24 0.0000334 1.629751e+07 0.0001691
AtBat 25 0.0000441 1.232847e+07 0.0002236
AtBat 26 0.0000584 9.326033e+06 0.0002955
AtBat 27 0.0000771 7.054802e+06 0.0003906
AtBat 28 0.0001020 5.336699e+06 0.0005162
AtBat 29 0.0001348 4.037017e+06 0.0006822
AtBat 30 0.0001781 3.053856e+06 0.0009015
AtBat 31 0.0002354 2.310130e+06 0.0011912
AtBat 32 0.0003110 1.747528e+06 0.0015737
AtBat 33 0.0004110 1.321941e+06 0.0020786
AtBat 34 0.0005430 1.000000e+06 0.0027448
AtBat 35 0.0007172 7.564633e+05 0.0036233
AtBat 36 0.0009471 5.722368e+05 0.0047807
AtBat 37 0.0012502 4.328761e+05 0.0063041
AtBat 38 0.0016496 3.274549e+05 0.0083062
AtBat 39 0.0021753 2.477076e+05 0.0109327
AtBat 40 0.0028562 1.873817e+05 0.0143452
AtBat 41 0.0037554 1.417474e+05 0.0188114
AtBat 42 0.0049293 1.072267e+05 0.0246060
AtBat 43 0.0064558 8.111308e+04 0.0320807
AtBat 44 0.0084309 6.135907e+04 0.0416500
AtBat 45 0.0109694 4.641589e+04 0.0537819
AtBat 46 0.0142045 3.511192e+04 0.0689717
AtBat 47 0.0182829 2.656088e+04 0.0876909
AtBat 48 0.0233542 2.009233e+04 0.1103053
AtBat 49 0.0295476 1.519911e+04 0.1369603
AtBat 50 0.0369572 1.149757e+04 0.1674595
AtBat 51 0.0455897 8.697490e+03 0.2011472
AtBat 52 0.0553341 6.579332e+03 0.2368793
AtBat 53 0.0659234 4.977024e+03 0.2731090
AtBat 54 0.0769203 3.764936e+03 0.3081306
AtBat 55 0.0877291 2.848036e+03 0.3403876
AtBat 56 0.0976389 2.154435e+03 0.3687727
AtBat 57 0.1058832 1.629751e+03 0.3927800
AtBat 58 0.1115181 1.232847e+03 0.4125082
AtBat 59 0.1139551 9.326033e+02 0.4284125
AtBat 60 0.1121111 7.054802e+02 0.4412059
AtBat 61 0.1052545 5.336699e+02 0.4515636
AtBat 62 0.0923834 4.037017e+02 0.4601078
AtBat 63 0.0721453 3.053856e+02 0.4673437
AtBat 64 0.0433386 2.310130e+02 0.4737111
AtBat 65 0.0038599 1.747528e+02 0.4794882
AtBat 66 -0.0480137 1.321941e+02 0.4849761
AtBat 67 -0.1143526 1.000000e+02 0.4903658
AtBat 68 -0.1969965 7.564633e+01 0.4957413
AtBat 69 -0.2970743 5.722368e+01 0.5011443
AtBat 70 -0.4148060 4.328761e+01 0.5065677
AtBat 71 -0.5489118 3.274549e+01 0.5119093
AtBat 72 -0.6964360 2.477076e+01 0.5170544
AtBat 73 -0.8532130 1.873817e+01 0.5219007
AtBat 74 -1.0135354 1.417474e+01 0.5263263
AtBat 75 -1.1712169 1.072267e+01 0.5302523
AtBat 76 -1.3203039 8.111308e+00 0.5336375
AtBat 77 -1.4557391 6.135907e+00 0.5364782
AtBat 78 -1.5737685 4.641589e+00 0.5388070
AtBat 79 -1.6728464 3.511192e+00 0.5406688
AtBat 80 -1.7526436 2.656088e+00 0.5421321
AtBat 81 -1.8153438 2.009233e+00 0.5432400
AtBat 82 -1.8619922 1.519911e+00 0.5440843
AtBat 83 -1.8962304 1.149757e+00 0.5446980
AtBat 84 -1.9201291 8.697490e-01 0.5451444
AtBat 85 -1.9368385 6.579332e-01 0.5454543
AtBat 86 -1.9482895 4.977024e-01 0.5456662
AtBat 87 -1.9562902 3.764936e-01 0.5458062
AtBat 88 -1.9617909 2.848036e-01 0.5458983
AtBat 89 -1.9654629 2.154435e-01 0.5459591
AtBat 90 -1.9680069 1.629751e-01 0.5459979
AtBat 91 -1.9697620 1.232847e-01 0.5460232
AtBat 92 -1.9709892 9.326030e-02 0.5460397
AtBat 93 -1.9718803 7.054800e-02 0.5460503
AtBat 94 -1.9725262 5.336700e-02 0.5460580
AtBat 95 -1.9729623 4.037020e-02 0.5460629
AtBat 96 -1.9732951 3.053860e-02 0.5460666
AtBat 97 -1.9735097 2.310130e-02 0.5460692
AtBat 98 -1.9736660 1.747530e-02 0.5460709
AtBat 99 -1.9737902 1.321940e-02 0.5460723
AtBat 100 -1.9738615 1.000000e-02 0.5460731
Hits 1 0.0000002 1.000000e+10 0.0000003
Hits 2 0.0000003 7.564633e+09 0.0000004
Hits 3 0.0000003 5.722368e+09 0.0000005
Hits 4 0.0000005 4.328761e+09 0.0000006
Hits 5 0.0000006 3.274549e+09 0.0000008
Hits 6 0.0000008 2.477076e+09 0.0000011
Hits 7 0.0000011 1.873817e+09 0.0000015
Hits 8 0.0000014 1.417474e+09 0.0000019
Hits 9 0.0000018 1.072267e+09 0.0000026
Hits 10 0.0000024 8.111308e+08 0.0000034
Hits 11 0.0000032 6.135907e+08 0.0000045
Hits 12 0.0000043 4.641589e+08 0.0000059
Hits 13 0.0000056 3.511192e+08 0.0000079
Hits 14 0.0000074 2.656088e+08 0.0000104
Hits 15 0.0000098 2.009233e+08 0.0000137
Hits 16 0.0000130 1.519911e+08 0.0000181
Hits 17 0.0000172 1.149757e+08 0.0000240
Hits 18 0.0000227 8.697490e+07 0.0000317
Hits 19 0.0000300 6.579332e+07 0.0000419
Hits 20 0.0000397 4.977024e+07 0.0000554
Hits 21 0.0000524 3.764936e+07 0.0000732
Hits 22 0.0000693 2.848036e+07 0.0000968
Hits 23 0.0000916 2.154435e+07 0.0001279
Hits 24 0.0001211 1.629751e+07 0.0001691
Hits 25 0.0001601 1.232847e+07 0.0002236
Hits 26 0.0002117 9.326033e+06 0.0002955
Hits 27 0.0002798 7.054802e+06 0.0003906
Hits 28 0.0003698 5.336699e+06 0.0005162
Hits 29 0.0004888 4.037017e+06 0.0006822
Hits 30 0.0006461 3.053856e+06 0.0009015
Hits 31 0.0008539 2.310130e+06 0.0011912
Hits 32 0.0011284 1.747528e+06 0.0015737
Hits 33 0.0014910 1.321941e+06 0.0020786
Hits 34 0.0019699 1.000000e+06 0.0027448
Hits 35 0.0026022 7.564633e+05 0.0036233
Hits 36 0.0034365 5.722368e+05 0.0047807
Hits 37 0.0045368 4.328761e+05 0.0063041
Hits 38 0.0059869 3.274549e+05 0.0083062
Hits 39 0.0078961 2.477076e+05 0.0109327
Hits 40 0.0103801 1.873817e+05 0.0143452
Hits 41 0.0136562 1.417474e+05 0.0188114
Hits 42 0.0179393 1.072267e+05 0.0246060
Hits 43 0.0235197 8.111308e+04 0.0320807
Hits 44 0.0307578 6.135907e+04 0.0416500
Hits 45 0.0400924 4.641589e+04 0.0537819
Hits 46 0.0520423 3.511192e+04 0.0689717
Hits 47 0.0671984 2.656088e+04 0.0876909
Hits 48 0.0861985 2.009233e+04 0.1103053
Hits 49 0.1096700 1.519911e+04 0.1369603
Hits 50 0.1381803 1.149757e+04 0.1674595
Hits 51 0.1721089 8.697490e+03 0.2011472
Hits 52 0.2115628 6.579332e+03 0.2368793
Hits 53 0.2562824 4.977024e+03 0.2731090
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Walks 16 0.0000273 1.519911e+08 0.0000181
Walks 17 0.0000361 1.149757e+08 0.0000240
Walks 18 0.0000477 8.697490e+07 0.0000317
Walks 19 0.0000631 6.579332e+07 0.0000419
Walks 20 0.0000834 4.977024e+07 0.0000554
Walks 21 0.0001103 3.764936e+07 0.0000732
Walks 22 0.0001457 2.848036e+07 0.0000968
Walks 23 0.0001927 2.154435e+07 0.0001279
Walks 24 0.0002547 1.629751e+07 0.0001691
Walks 25 0.0003367 1.232847e+07 0.0002236
Walks 26 0.0004450 9.326033e+06 0.0002955
Walks 27 0.0005882 7.054802e+06 0.0003906
Walks 28 0.0007775 5.336699e+06 0.0005162
Walks 29 0.0010277 4.037017e+06 0.0006822
Walks 30 0.0013582 3.053856e+06 0.0009015
Walks 31 0.0017950 2.310130e+06 0.0011912
Walks 32 0.0023720 1.747528e+06 0.0015737
Walks 33 0.0031342 1.321941e+06 0.0020786
Walks 34 0.0041406 1.000000e+06 0.0027448
Walks 35 0.0054690 7.564633e+05 0.0036233
Walks 36 0.0072218 5.722368e+05 0.0047807
Walks 37 0.0095329 4.328761e+05 0.0063041
Walks 38 0.0125778 3.274549e+05 0.0083062
Walks 39 0.0165851 2.477076e+05 0.0109327
Walks 40 0.0218179 1.873817e+05 0.0143452
Walks 41 0.0287020 1.417474e+05 0.0188114
Walks 42 0.0377005 1.072267e+05 0.0246060
Walks 43 0.0494220 8.111308e+04 0.0320807
Walks 44 0.0646214 6.135907e+04 0.0416500
Walks 45 0.0842158 4.641589e+04 0.0537819
Walks 46 0.1092880 3.511192e+04 0.0689717
Walks 47 0.1410672 2.656088e+04 0.0876909
Walks 48 0.1808738 2.009233e+04 0.1103053
Walks 49 0.2300125 1.519911e+04 0.1369603
Walks 50 0.2896187 1.149757e+04 0.1674595
Walks 51 0.3604422 8.697490e+03 0.2011472
Walks 52 0.4426320 6.579332e+03 0.2368793
Walks 53 0.5355508 4.977024e+03 0.2731090
Walks 54 0.6377760 3.764936e+03 0.3081306
Walks 55 0.7471760 2.848036e+03 0.3403876
Walks 56 0.8612172 2.154435e+03 0.3687727
Walks 57 0.9772875 1.629751e+03 0.3927800
Walks 58 1.0934418 1.232847e+03 0.4125082
Walks 59 1.2076718 9.326033e+02 0.4284125
Walks 60 1.3198795 7.054802e+02 0.4412059
Walks 61 1.4298316 5.336699e+02 0.4515636
Walks 62 1.5386280 4.037017e+02 0.4601078
Walks 63 1.6490380 3.053856e+02 0.4673437
Walks 64 1.7635487 2.310130e+02 0.4737111
Walks 65 1.8878871 1.747528e+02 0.4794882
Walks 66 2.0268310 1.321941e+02 0.4849761
Walks 67 2.1866738 1.000000e+02 0.4903658
Walks 68 2.3726576 7.564633e+01 0.4957413
Walks 69 2.5882307 5.722368e+01 0.5011443
Walks 70 2.8350843 4.328761e+01 0.5065677
Walks 71 3.1102968 3.274549e+01 0.5119093
Walks 72 3.4076321 2.477076e+01 0.5170544
Walks 73 3.7200425 1.873817e+01 0.5219007
Walks 74 4.0365649 1.417474e+01 0.5263263
Walks 75 4.3466515 1.072267e+01 0.5302523
Walks 76 4.6407848 8.111308e+00 0.5336375
Walks 77 4.9110792 6.135907e+00 0.5364782
Walks 78 5.1515962 4.641589e+00 0.5388070
Walks 79 5.3600785 3.511192e+00 0.5406688
Walks 80 5.5357165 2.656088e+00 0.5421321
Walks 81 5.6812245 2.009233e+00 0.5432400
Walks 82 5.7981554 1.519911e+00 0.5440843
Walks 83 5.8910979 1.149757e+00 0.5446980
Walks 84 5.9635746 8.697490e-01 0.5451444
Walks 85 6.0198809 6.579332e-01 0.5454543
Walks 86 6.0633585 4.977024e-01 0.5456662
Walks 87 6.0969219 3.764936e-01 0.5458062
Walks 88 6.1227027 2.848036e-01 0.5458983
Walks 89 6.1423731 2.154435e-01 0.5459591
Walks 90 6.1573465 1.629751e-01 0.5459979
Walks 91 6.1687194 1.232847e-01 0.5460232
Walks 92 6.1772886 9.326030e-02 0.5460397
Walks 93 6.1836867 7.054800e-02 0.5460503
Walks 94 6.1886119 5.336700e-02 0.5460580
Walks 95 6.1921207 4.037020e-02 0.5460629
Walks 96 6.1949605 3.053860e-02 0.5460666
Walks 97 6.1969498 2.310130e-02 0.5460692
Walks 98 6.1984355 1.747530e-02 0.5460709
Walks 99 6.1996526 1.321940e-02 0.5460723
Walks 100 6.2003772 1.000000e-02 0.5460731
Years 1 0.0000017 1.000000e+10 0.0000003
Years 2 0.0000022 7.564633e+09 0.0000004
Years 3 0.0000030 5.722368e+09 0.0000005
Years 4 0.0000039 4.328761e+09 0.0000006
Years 5 0.0000052 3.274549e+09 0.0000008
Years 6 0.0000069 2.477076e+09 0.0000011
Years 7 0.0000091 1.873817e+09 0.0000015
Years 8 0.0000120 1.417474e+09 0.0000019
Years 9 0.0000158 1.072267e+09 0.0000026
Years 10 0.0000209 8.111308e+08 0.0000034
Years 11 0.0000277 6.135907e+08 0.0000045
Years 12 0.0000366 4.641589e+08 0.0000059
Years 13 0.0000484 3.511192e+08 0.0000079
Years 14 0.0000639 2.656088e+08 0.0000104
Years 15 0.0000845 2.009233e+08 0.0000137
Years 16 0.0001117 1.519911e+08 0.0000181
Years 17 0.0001477 1.149757e+08 0.0000240
Years 18 0.0001952 8.697490e+07 0.0000317
Years 19 0.0002580 6.579332e+07 0.0000419
Years 20 0.0003411 4.977024e+07 0.0000554
Years 21 0.0004509 3.764936e+07 0.0000732
Years 22 0.0005960 2.848036e+07 0.0000968
Years 23 0.0007879 2.154435e+07 0.0001279
Years 24 0.0010415 1.629751e+07 0.0001691
Years 25 0.0013767 1.232847e+07 0.0002236
Years 26 0.0018198 9.326033e+06 0.0002955
Years 27 0.0024054 7.054802e+06 0.0003906
Years 28 0.0031794 5.336699e+06 0.0005162
Years 29 0.0042023 4.037017e+06 0.0006822
Years 30 0.0055538 3.053856e+06 0.0009015
Years 31 0.0073396 2.310130e+06 0.0011912
Years 32 0.0096984 1.747528e+06 0.0015737
Years 33 0.0128138 1.321941e+06 0.0020786
Years 34 0.0169268 1.000000e+06 0.0027448
Years 35 0.0223550 7.564633e+05 0.0036233
Years 36 0.0295147 5.722368e+05 0.0047807
Years 37 0.0389519 4.328761e+05 0.0063041
Years 38 0.0513791 3.274549e+05 0.0083062
Years 39 0.0677236 2.477076e+05 0.0109327
Years 40 0.0888350 1.873817e+05 0.0143452
Years 41 0.1167003 1.417474e+05 0.0188114
Years 42 0.1530045 1.072267e+05 0.0246060
Years 43 0.2000875 8.111308e+04 0.0320807
Years 44 0.2607876 6.135907e+04 0.0416500
Years 45 0.3384406 4.641589e+04 0.0537819
Years 46 0.4367954 3.511192e+04 0.0689717
Years 47 0.5597875 2.656088e+04 0.0876909
Years 48 0.7111047 2.009233e+04 0.1103053
Years 49 0.8934578 1.519911e+04 0.1369603
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Years 51 1.3515674 8.697490e+03 0.2011472
Years 52 1.6186210 6.579332e+03 0.2368793
Years 53 1.8973479 4.977024e+03 0.2731090
Years 54 2.1719538 3.764936e+03 0.3081306
Years 55 2.4227711 2.848036e+03 0.3403876
Years 56 2.6284632 2.154435e+03 0.3687727
Years 57 2.7672071 1.629751e+03 0.3927800
Years 58 2.8197715 1.232847e+03 0.4125082
Years 59 2.7669044 9.326033e+02 0.4284125
Years 60 2.5964042 7.054802e+02 0.4412059
Years 61 2.2906522 5.336699e+02 0.4515636
Years 62 1.8339550 4.037017e+02 0.4601078
Years 63 1.2252824 3.053856e+02 0.4673437
Years 64 0.4287021 2.310130e+02 0.4737111
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Years 98 -3.7460508 1.747530e-02 0.5460709
Years 99 -3.7260113 1.321940e-02 0.5460723
Years 100 -3.7140342 1.000000e-02 0.5460731
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CAtBat 2 0.0000000 7.564633e+09 0.0000004
CAtBat 3 0.0000000 5.722368e+09 0.0000005
CAtBat 4 0.0000000 4.328761e+09 0.0000006
CAtBat 5 0.0000000 3.274549e+09 0.0000008
CAtBat 6 0.0000000 2.477076e+09 0.0000011
CAtBat 7 0.0000000 1.873817e+09 0.0000015
CAtBat 8 0.0000000 1.417474e+09 0.0000019
CAtBat 9 0.0000000 1.072267e+09 0.0000026
CAtBat 10 0.0000001 8.111308e+08 0.0000034
CAtBat 11 0.0000001 6.135907e+08 0.0000045
CAtBat 12 0.0000001 4.641589e+08 0.0000059
CAtBat 13 0.0000001 3.511192e+08 0.0000079
CAtBat 14 0.0000002 2.656088e+08 0.0000104
CAtBat 15 0.0000002 2.009233e+08 0.0000137
CAtBat 16 0.0000003 1.519911e+08 0.0000181
CAtBat 17 0.0000004 1.149757e+08 0.0000240
CAtBat 18 0.0000005 8.697490e+07 0.0000317
CAtBat 19 0.0000007 6.579332e+07 0.0000419
CAtBat 20 0.0000009 4.977024e+07 0.0000554
CAtBat 21 0.0000012 3.764936e+07 0.0000732
CAtBat 22 0.0000016 2.848036e+07 0.0000968
CAtBat 23 0.0000022 2.154435e+07 0.0001279
CAtBat 24 0.0000029 1.629751e+07 0.0001691
CAtBat 25 0.0000038 1.232847e+07 0.0002236
CAtBat 26 0.0000050 9.326033e+06 0.0002955
CAtBat 27 0.0000066 7.054802e+06 0.0003906
CAtBat 28 0.0000088 5.336699e+06 0.0005162
CAtBat 29 0.0000116 4.037017e+06 0.0006822
CAtBat 30 0.0000153 3.053856e+06 0.0009015
CAtBat 31 0.0000202 2.310130e+06 0.0011912
CAtBat 32 0.0000267 1.747528e+06 0.0015737
CAtBat 33 0.0000353 1.321941e+06 0.0020786
CAtBat 34 0.0000466 1.000000e+06 0.0027448
CAtBat 35 0.0000615 7.564633e+05 0.0036233
CAtBat 36 0.0000813 5.722368e+05 0.0047807
CAtBat 37 0.0001073 4.328761e+05 0.0063041
CAtBat 38 0.0001415 3.274549e+05 0.0083062
CAtBat 39 0.0001865 2.477076e+05 0.0109327
CAtBat 40 0.0002450 1.873817e+05 0.0143452
CAtBat 41 0.0003220 1.417474e+05 0.0188114
CAtBat 42 0.0004224 1.072267e+05 0.0246060
CAtBat 43 0.0005530 8.111308e+04 0.0320807
CAtBat 44 0.0007216 6.135907e+04 0.0416500
CAtBat 45 0.0009380 4.641589e+04 0.0537819
CAtBat 46 0.0012132 3.511192e+04 0.0689717
CAtBat 47 0.0015592 2.656088e+04 0.0876909
CAtBat 48 0.0019880 2.009233e+04 0.1103053
CAtBat 49 0.0025100 1.519911e+04 0.1369603
CAtBat 50 0.0031318 1.149757e+04 0.1674595
CAtBat 51 0.0038533 8.697490e+03 0.2011472
CAtBat 52 0.0046653 6.579332e+03 0.2368793
CAtBat 53 0.0055475 4.977024e+03 0.2731090
CAtBat 54 0.0064699 3.764936e+03 0.3081306
CAtBat 55 0.0073947 2.848036e+03 0.3403876
CAtBat 56 0.0082818 2.154435e+03 0.3687727
CAtBat 57 0.0090920 1.629751e+03 0.3927800
CAtBat 58 0.0098042 1.232847e+03 0.4125082
CAtBat 59 0.0103811 9.326033e+02 0.4284125
CAtBat 60 0.0108341 7.054802e+02 0.4412059
CAtBat 61 0.0111426 5.336699e+02 0.4515636
CAtBat 62 0.0112975 4.037017e+02 0.4601078
CAtBat 63 0.0113216 3.053856e+02 0.4673437
CAtBat 64 0.0111327 2.310130e+02 0.4737111
CAtBat 65 0.0108125 1.747528e+02 0.4794882
CAtBat 66 0.0102502 1.321941e+02 0.4849761
CAtBat 67 0.0093988 1.000000e+02 0.4903658
CAtBat 68 0.0082189 7.564633e+01 0.4957413
CAtBat 69 0.0065957 5.722368e+01 0.5011443
CAtBat 70 0.0043812 4.328761e+01 0.5065677
CAtBat 71 0.0014293 3.274549e+01 0.5119093
CAtBat 72 -0.0024658 2.477076e+01 0.5170544
CAtBat 73 -0.0074850 1.873817e+01 0.5219007
CAtBat 74 -0.0138037 1.417474e+01 0.5263263
CAtBat 75 -0.0215482 1.072267e+01 0.5302523
CAtBat 76 -0.0307521 8.111308e+00 0.5336375
CAtBat 77 -0.0413482 6.135907e+00 0.5364782
CAtBat 78 -0.0532062 4.641589e+00 0.5388070
CAtBat 79 -0.0659320 3.511192e+00 0.5406688
CAtBat 80 -0.0792321 2.656088e+00 0.5421321
CAtBat 81 -0.0922939 2.009233e+00 0.5432400
CAtBat 82 -0.1052349 1.519911e+00 0.5440843
CAtBat 83 -0.1171587 1.149757e+00 0.5446980
CAtBat 84 -0.1281349 8.697490e-01 0.5451444
CAtBat 85 -0.1376330 6.579332e-01 0.5454543
CAtBat 86 -0.1457004 4.977024e-01 0.5456662
CAtBat 87 -0.1522684 3.764936e-01 0.5458062
CAtBat 88 -0.1575691 2.848036e-01 0.5458983
CAtBat 89 -0.1618281 2.154435e-01 0.5459591
CAtBat 90 -0.1650985 1.629751e-01 0.5459979
CAtBat 91 -0.1676366 1.232847e-01 0.5460232
CAtBat 92 -0.1695734 9.326030e-02 0.5460397
CAtBat 93 -0.1710180 7.054800e-02 0.5460503
CAtBat 94 -0.1721853 5.336700e-02 0.5460580
CAtBat 95 -0.1730196 4.037020e-02 0.5460629
CAtBat 96 -0.1737195 3.053860e-02 0.5460666
CAtBat 97 -0.1742244 2.310130e-02 0.5460692
CAtBat 98 -0.1746012 1.747530e-02 0.5460709
CAtBat 99 -0.1749136 1.321940e-02 0.5460723
CAtBat 100 -0.1751006 1.000000e-02 0.5460731
CHits 1 0.0000000 1.000000e+10 0.0000003
CHits 2 0.0000000 7.564633e+09 0.0000004
CHits 3 0.0000000 5.722368e+09 0.0000005
CHits 4 0.0000000 4.328761e+09 0.0000006
CHits 5 0.0000001 3.274549e+09 0.0000008
CHits 6 0.0000001 2.477076e+09 0.0000011
CHits 7 0.0000001 1.873817e+09 0.0000015
CHits 8 0.0000001 1.417474e+09 0.0000019
CHits 9 0.0000002 1.072267e+09 0.0000026
CHits 10 0.0000002 8.111308e+08 0.0000034
CHits 11 0.0000003 6.135907e+08 0.0000045
CHits 12 0.0000004 4.641589e+08 0.0000059
CHits 13 0.0000005 3.511192e+08 0.0000079
CHits 14 0.0000006 2.656088e+08 0.0000104
CHits 15 0.0000009 2.009233e+08 0.0000137
CHits 16 0.0000011 1.519911e+08 0.0000181
CHits 17 0.0000015 1.149757e+08 0.0000240
CHits 18 0.0000020 8.697490e+07 0.0000317
CHits 19 0.0000026 6.579332e+07 0.0000419
CHits 20 0.0000035 4.977024e+07 0.0000554
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CHits 22 0.0000060 2.848036e+07 0.0000968
CHits 23 0.0000080 2.154435e+07 0.0001279
CHits 24 0.0000106 1.629751e+07 0.0001691
CHits 25 0.0000139 1.232847e+07 0.0002236
CHits 26 0.0000184 9.326033e+06 0.0002955
CHits 27 0.0000244 7.054802e+06 0.0003906
CHits 28 0.0000322 5.336699e+06 0.0005162
CHits 29 0.0000426 4.037017e+06 0.0006822
CHits 30 0.0000563 3.053856e+06 0.0009015
CHits 31 0.0000744 2.310130e+06 0.0011912
CHits 32 0.0000983 1.747528e+06 0.0015737
CHits 33 0.0001298 1.321941e+06 0.0020786
CHits 34 0.0001715 1.000000e+06 0.0027448
CHits 35 0.0002265 7.564633e+05 0.0036233
CHits 36 0.0002991 5.722368e+05 0.0047807
CHits 37 0.0003947 4.328761e+05 0.0063041
CHits 38 0.0005207 3.274549e+05 0.0083062
CHits 39 0.0006864 2.477076e+05 0.0109327
CHits 40 0.0009022 1.873817e+05 0.0143452
CHits 41 0.0011861 1.417474e+05 0.0188114
CHits 42 0.0015566 1.072267e+05 0.0246060
CHits 43 0.0020383 8.111308e+04 0.0320807
CHits 44 0.0026612 6.135907e+04 0.0416500
CHits 45 0.0034616 4.641589e+04 0.0537819
CHits 46 0.0044811 3.511192e+04 0.0689717
CHits 47 0.0057658 2.656088e+04 0.0876909
CHits 48 0.0073628 2.009233e+04 0.1103053
CHits 49 0.0093151 1.519911e+04 0.1369603
CHits 50 0.0116536 1.149757e+04 0.1674595
CHits 51 0.0143886 8.697490e+03 0.2011472
CHits 52 0.0175004 6.579332e+03 0.2368793
CHits 53 0.0209351 4.977024e+03 0.2731090
CHits 54 0.0246075 3.764936e+03 0.3081306
CHits 55 0.0284119 2.848036e+03 0.3403876
CHits 56 0.0322403 2.154435e+03 0.3687727
CHits 57 0.0360011 1.629751e+03 0.3927800
CHits 58 0.0396794 1.232847e+03 0.4125082
CHits 59 0.0432193 9.326033e+02 0.4284125
CHits 60 0.0467456 7.054802e+02 0.4412059
CHits 61 0.0503162 5.336699e+02 0.4515636
CHits 62 0.0540663 4.037017e+02 0.4601078
CHits 63 0.0581936 3.053856e+02 0.4673437
CHits 64 0.0628490 2.310130e+02 0.4737111
CHits 65 0.0682373 1.747528e+02 0.4794882
CHits 66 0.0745605 1.321941e+02 0.4849761
CHits 67 0.0820561 1.000000e+02 0.4903658
CHits 68 0.0907726 7.564633e+01 0.4957413
CHits 69 0.1006909 5.722368e+01 0.5011443
CHits 70 0.1118777 4.328761e+01 0.5065677
CHits 71 0.1239214 3.274549e+01 0.5119093
CHits 72 0.1362730 2.477076e+01 0.5170544
CHits 73 0.1489150 1.873817e+01 0.5219007
CHits 74 0.1611629 1.417474e+01 0.5263263
CHits 75 0.1726364 1.072267e+01 0.5302523
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Exercise Ridge Regression

Use the birthwt data. Your response is variable bwt.

Apply a ridge regression.

Then create training and testing set(split your preference).

Create two charts:

  • Chart1 : X-axis lambda value, Y-axis MSE. Plot train and test MSE for each lambda value.
  • Chart2: X-axis lambda value, Y-axis standardized coeffs. Plot coeff estimates for all predictors.